Highly efficient and exact method for parallelization of grid‐based algorithms and its implementation in DelPhi

The Gauss–Seidel (GS) method is a standard iterative numerical method widely used to solve a system of equations and, in general, is more efficient comparing to other iterative methods, such as the Jacobi method. However, standard implementation of the GS method restricts its utilization in parallel computing due to its requirement of using updated neighboring values (i.e., in current iteration) as soon as they are available. Here, we report an efficient and exact (not requiring assumptions) method to parallelize iterations and to reduce the computational time as a linear/nearly linear function of the number of processes or computing units. In contrast to other existing solutions, our method does not require any assumptions and is equally applicable for solving linear and nonlinear equations. This approach is implemented in the DelPhi program, which is a finite difference Poisson–Boltzmann equation solver to model electrostatics in molecular biology. This development makes the iterative procedure on obtaining the electrostatic potential distribution in the parallelized DelPhi several folds faster than that in the serial code. Further, we demonstrate the advantages of the new parallelized DelPhi by computing the electrostatic potential and the corresponding energies of large supramolecular structures. © 2012 Wiley Periodicals, Inc.     

Comment on the validation of continuum electrostatics models

A validation based on solvation energies (vacuum to water transfer) is not sufficient to justify the use of approximated models of electrostatics to rank ligand/protein complexes. A full validation should be based on energies in solution, i.e., solvation plus vacuum Coulomb energies, because of the anticorrelation between solvation and vacuum energies. The energy in solution is the relevant quantity in simulations of biological macromolecules and complexes. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1533–1536, 1999     

Statistical investigation of surface bound ions and further development of BION server to include pH and salt dependence

Ions are engaged in multiple biological processes in cells. By binding to the macromolecules or being mobile in the solvent, they maintain the integrity of the structure of macromolecules; participate in their enzymatic activity; or screen electrostatic interactions. While experimental methods are not always able to assign the exact location of ions, computational methods are in demand. Although the majority of computational methods are successful in predicting the position of ions buried inside macromolecules, they are less effective in deciphering positions of surface bound ions. Here, we propose the new BION algorithm ( http://compbio.clemson.edu/bion_server_ph/ ) that predicts the location of the surface bound ions. It is more efficient and accurate compared to the previous version since it uses more advanced clustering algorithm in combination with pairing rules. In addition, the BION webserver allows specifying the pH and the salt concentration in predicting ions positions. © 2015 Wiley Periodicals, Inc.     

O(NlogN) continuous electrostatics method for fast calculation of solvation energies of biomolecules

We report the development of a new, fast, surface-based method for numerical calculations of the solvation energy of biomolecules with a large number of charged groups. The procedure scales linearly with the system size, both in time and memory requirements, produces explicit values for the reaction field potential and stable values of the polar energy within only a few percent error margin practically for any molecular configurations. The method works well both for large and small molecules and thus gives stable energy differences for quantities such as the polar energy contributions to molecular complex formation energies.     

Hybrid boundary element and finite difference method for solving the nonlinear Poisson-Boltzmann equation

A hybrid approach for solving the nonlinear Poisson-Boltzmann equation (PBE) is presented. Under this approach, the electrostatic potential is separated into (1) a linear component satisfying the linear PBE and solved using a fast boundary element method and (2) a correction term accounting for nonlinear effects and optionally, the presence of an ion-exclusion layer. Because the correction potential contains no singularities (in particular, it is smooth at charge sites) it can be accurately and efficiently solved using a finite difference method. The motivation for and formulation of such a decomposition are presented together with the numerical method for calculating the linear and correction potentials. For comparison, we also develop an integral equation representation of the solution to the nonlinear PBE. When implemented upon regular lattice grids, the hybrid scheme is found to outperform the integral equation method when treating nonlinear PBE problems. Results are presented for a spherical cavity containing a central charge, where the objective is to compare computed 1D nonlinear PBE solutions against ones obtained with alternate numerical solution methods. This is followed by examination of the electrostatic properties of nucleic acid structures.     

Application of the thin-shell formulation to the numerical modeling of Stern layer in biomolecular electrostatics

In this article, the thin-shell formulation is applied to efficiently modeling the Stern layer within computational algorithms oriented toward the boundary element solution of the linearized Poisson-Boltzmann equation. The attention is focused on the calculation of the electrostatic potential in proximity to a biomolecule immersed in an electrolyte medium. Following the proposed approach, the Stern layer is made to collapse to a zero-thickness region (two-dimensional surface) with interface conditions linking the electrostatic potential over the molecular and the bulk ion accessible surfaces. Advantages lie in the limitation of divergent integral problems and in the halving of the unknown number, with a significant impact on computational time and memory requirements when modeling large biomolecules.     

A new massively parallel version of CRYSTAL for large systems on high performance computing architectures

Fully ab initio treatment of complex solid systems needs computational software which is able to efficiently take advantage of the growing power of high performance computing (HPC) architectures. Recent improvements in CRYSTAL, a periodic ab initio code that uses a Gaussian basis set, allows treatment of very large unit cells for crystalline systems on HPC architectures with high parallel efficiency in terms of running time and memory requirements. The latter is a crucial point, due to the trend toward architectures relying on a very high number of cores with associated relatively low memory availability. An exhaustive performance analysis shows that density functional calculations, based on a hybrid functional, of low‐symmetry systems containing up to 100,000 atomic orbitals and 8000 atoms are feasible on the most advanced HPC architectures available to European researchers today, using thousands of processors. © 2012 Wiley Periodicals, Inc.     

An Approximate Expression for the Degree of Counterion Binding of Dressed Micelles

A simple formula for calculating the degree of counterion binding of dressed micelles is presented. This approximate expression for the spherical micelle is derived from the extending Langmuir's method in the case of high surface potentials. It works quite well for the estimation of the degree of counterion binding of micelles. The simple form is very convenient for practical use.     

PRIRODA-04: a quantum-chemical program suite. New possibilities in the study of molecular systems with the application of parallel computing

Main characteristics are described of the PRIRODA quantum-chemical program suite designed for the study of complex molecular systems by the density functional theory, at the MP2, MP3, and MP4 levels of multiparticle perturbation theory, and by the coupled-cluster single and double excitations method (CCSD) with the application of parallel computing. A number of examples of calculations are presented.__________Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 3, pp. 804–810, March, 2005.     

A scalable parallel Monte Carlo method for free energy simulations of molecular systems

We present a method of parallelizing flat histogram Monte Carlo simulations, which give the free energy of a molecular system as an output. In the serial version, a constant probability distribution, as a function of any system parameter, is calculated by updating an external potential that is added to the system Hamiltonian. This external potential is related to the free energy. In the parallel implementation, the simulation is distributed on to different processors. With regular intervals the modifying potential is summed over all processors and distributed back to every processor, thus spreading the information of which parts of parameter space have been explored. This implementation is shown to decrease the execution time linearly with added number of processors.     

On the irrelevance of electrostatics for the crystal structures and polymorphism of long even n-alkanes

It is known that the experimental triclinic crystal structures of even n-alkanes are not well reproduced upon energy minimization with current force fields. The inclusion of electrostatics does not solve this, and, moreover, some charge schemes show unphysical features such as positively charged carbon atoms or charge alternation. The effect of the electrostatics on the energies of the crystal structures of the even n-alkanes, and thereby on their polymorphism, has never been established. A new charge scheme is introduced that yields physically sensible charges without constraints. It will also be shown, however, that electrostatics are relevant neither for the structures of the crystals, nor for their energies.     

Dielectric relaxation in proteins: a continuum electrostatics model incorporating dielectric heterogeneity of the protein and time‐dependent charges

A boundary element formulation of continuum electrostatics is used to examine time‐independent dielectric relaxation and screening in two proteins, and time‐dependent relaxation in two simpler solutes. Cytochrome c oxidation is modeled by inserting partial charges on the heme, using one to three dielectric regions in the protein. It was suggested recently that for charge insertion on a protein‐bound ligand, all or part of the ligand should be treated as a cavity within the protein medium. Here, the effect of an internal cavity surrounding the central heme atoms is examined, considering separately the static and relaxation (or reorganization) free energies. The former is the free energy to remove the redox electron while maintaining the rest of the structure and charge distribution fixed; the latter is the free energy associated with the relaxation into the product state after the corresponding constraints are released. The effect of the cavity is found to be small for the static free energy, while for the relaxation free energy it is large, as polarization of groups immediately around the heme dominates the relaxation. If the protein surface groups are treated as a distinct medium with a dielectric of 25 (as suggested by recent molecular dynamics simulations), the relaxation free energy decreases significantly (from −37.0 to −43.9 kcal/mol), compared to a model where the whole protein has a dielectric constant of two. Therefore, with this model, although polarization of groups immediately around the heme still dominates the relaxation, polar groups near the protein surface also contribute significantly, and solvent negligibly. The screening of an applied field within myoglobin is calculated, with the protein surrounded by either a low‐dielectric or a high‐dielectric glass. In the vicinity of the CO ligand, the screening is approximately isotropic with a low‐dielectric glass. It is anisotropic with a high‐dielectric glass, but the applied and local fields are still approximately parallel. This has implications for experiments that probe dielectric screening in proteins with the newly developed technique of vibrational Stark spectroscopy: with a high‐dielectric glass, a single, rotationally averaged screening factor can be used, the local field being about 1.65 times the applied field. Finally, we calculate the time‐dependent relaxation in response to instantaneous charge insertion within a spherical cavity in a Debye solvent, and to photoexcitation of a tryptophan solute, illustrating the extension of the boundary element formulation to time‐dependent problems. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 290–305, 2001     

Modeling the electrostatic potential of asymmetric lipopolysaccharide membranes: The MEMPOT algorithm implemented in DelPhi

Four chemotypes of the rough lipopolysaccharides (LPS) membrane from Pseudomonas aeruginosa were investigated by a combined approach of explicit water molecular dynamics (MD) simulations and Poisson–Boltzmann continuum electrostatics with the goal to deliver the distribution of the electrostatic potential across the membrane. For the purpose of this investigation, a new tool for modeling the electrostatic potential profile along the axis normal to the membrane, MEMbrane POTential (MEMPOT), was developed and implemented in DelPhi. Applying MEMPOT on the snapshots obtained by MD simulations, two observations were made: (a) the average electrostatic potential has a complex profile but is mostly positive inside the membrane due to the presence of Ca²⁺ ions, which overcompensate for the negative potential created by lipid phosphate groups; and (b) correct modeling of the electrostatic potential profile across the membrane requires taking into account the water phase, while neglecting it (vacuum calculations) results in dramatic changes including a reversal of the sign of the potential inside the membrane. Furthermore, using DelPhi to assign different dielectric constants for different regions of the LPS membranes, it was investigated whether a single frame structure before MD simulations with appropriate dielectric constants for the lipid tails, inner, and the external leaflet regions, can deliver the same average electrostatic potential distribution as obtained from the MD‐generated ensemble of structures. Indeed, this can be attained by using smaller dielectric constant for the tail and inner leaflet regions (mostly hydrophobic) than for the external leaflet region (hydrophilic) and the optimal dielectric constant values are chemotype‐specific. © 2014 Wiley Periodicals, Inc.     

Constructing irregular surfaces to enclose macromolecular complexes for mesoscale modeling using the discrete surface charge optimization (DISCO) algorithm

Salt-mediated electrostatics interactions play an essential role in biomolecular structures and dynamics. Because macromolecular systems modeled at atomic resolution contain thousands of solute atoms, the electrostatic computations constitute an expensive part of the force and energy calculations. Implicit solvent models are one way to simplify the model and associated calculations, but they are generally used in combination with standard atomic models for the solute. To approximate electrostatics interactions in models on the polymer level (e.g., supercoiled DNA) that are simulated over long times (e.g., milliseconds) using Brownian dynamics, Beard and Schlick have developed the DiSCO (Discrete Surface Charge Optimization) algorithm. DiSCO represents a macromolecular complex by a few hundred discrete charges on a surface enclosing the system modeled by the Debye-Hückel (screened Coulombic) approximation to the Poisson-Boltzmann equation, and treats the salt solution as continuum solvation. DiSCO can represent the nucleosome core particle (>12,000 atoms), for example, by 353 discrete surface charges distributed on the surfaces of a large disk for the nucleosome core particle and a slender cylinder for the histone tail; the charges are optimized with respect to the Poisson-Boltzmann solution for the electric field, yielding a approximately 5.5% residual. Because regular surfaces enclosing macromolecules are not sufficiently general and may be suboptimal for certain systems, we develop a general method to construct irregular models tailored to the geometry of macromolecules. We also compare charge optimization based on both the electric field and electrostatic potential refinement. Results indicate that irregular surfaces can lead to a more accurate approximation (lower residuals), and the refinement in terms of the electric field is more robust. We also show that surface smoothing for irregular models is important, that the charge optimization (by the TNPACK minimizer) is efficient and does not depend on the initial assigned values, and that the residual is acceptable when the distance to the model surface is close to, or larger than, the Debye length. We illustrate applications of DiSCO's model-building procedure to chromatin folding and supercoiled DNA bound to Hin and Fis proteins. DiSCO is generally applicable to other interesting macromolecular systems for which mesoscale models are appropriate, to yield a resolution between the all-atom representative and the polymer level.     

The Calculation of Surface Potential for Spherical Ionic Micelles

In this article, an approximate expression for the calculation of surface potential for spherical ionic micelles has been presented. This simple analytic form overcomes the complexity of original theory. The calculated values of surface potential of spherical ionic micelles are in quite good agreement with the exact numerical values of nonlinear Poisson‐Boltzmann equation.     

A fast, scalable method for the parallel evaluation of distance-limited pairwise particle interactions

Classical molecular dynamics simulations of biological macromolecules in explicitly modeled solvent typically require the evaluation of interactions between all pairs of atoms separated by no more than some distance R, with more distant interactions handled using some less expensive method. Performing such simulations for periods on the order of a millisecond is likely to require the use of massive parallelism. The extent to which such simulations can be efficiently parallelized, however, has historically been limited by the time required for interprocessor communication. This article introduces a new method for the parallel evaluation of distance-limited pairwise particle interactions that significantly reduces the amount of data transferred between processors by comparison with traditional methods. Specifically, the amount of data transferred into and out of a given processor scales as O(R(3/2)p(-1/2)), where p is the number of processors, and with constant factors that should yield a substantial performance advantage in practice.     

Ultrafast vibrational spectroscopy and relaxation in polyatomic molecules: Potential for molecular parallel computing

The feasibility of controlled ultrafast pumping in the mid IR and the probe of the subsequent intramolecular dynamics is illustrated for vibrational excitation of the two metal carbonyls W(CO)₆ and Mn(CO)₅Br in solution. Pumping and probing is performed by short, 130 fs, pulses centered at about 2000 cm⁻¹. Frequency resolved measurements of the time delayed probe pulse are performed. Measured two dimensional spectra are fitted by a kinetic scheme that models the vibrational dynamics. Fast relaxation is solvent induced with the solvent acting also as a heat bath. The (several) probe signals in the experiment can be thought of as the response of a finite state logic machine. This suggests that the molecular machine can act as an ultrafast (petaHertz) processor. The number of internal (memory) states of the machine is determined by the number of vibrational states in the kinetic scheme that can fit the observed relaxation. The number of outputs of the machine is the number of the several different available probe signals. It is shown that the machine is massively parallel because in each (sub ps) time step it produces an entire vector as an output and that each component of the output vector is, by itself, a transform over the input. Beyond that, the machine can produce a (finite number of) different output vectors in sequential time steps.     

Adaptive multilevel finite element solution of the Poisson–Boltzmann equation II. Refinement at solvent‐accessible surfaces in biomolecular systems

We apply the adaptive multilevel finite element techniques (Holst, Baker, and Wang 21 ) to the nonlinear Poisson–Boltzmann equation (PBE) in the context of biomolecules. Fast and accurate numerical solution of the PBE in this setting is usually difficult to accomplish due to presence of discontinuous coefficients, delta functions, three spatial dimensions, unbounded domains, and rapid (exponential) nonlinearity. However, these adaptive techniques have shown substantial improvement in solution time over conventional uniform‐mesh finite difference methods. One important aspect of the adaptive multilevel finite element method is the robust a posteriori error estimators necessary to drive the adaptive refinement routines. This article discusses the choice of solvent accessibility for a posteriori error estimation of PBE solutions and the implementation of such routines in the “Adaptive Poisson–Boltzmann Solver” (APBS) software package based on the “Manifold Code” (MC) libraries. Results are shown for the application of this method to several biomolecular systems. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1343–1352, 2000     

irGPU.proton.Net: Irregular strong charge interaction networks of protonatable groups in protein molecules—a GPU solver using the fast multipole method and statistical thermodynamics

Motivation: A cluster of strongly interacting ionization groups in protein molecules with irregular ionization behavior is suggestive for specific structure–function relationship. However, their computational treatment is unconventional (e.g., lack of convergence in naive self‐consistent iterative algorithm). The stringent evaluation requires evaluation of Boltzmann averaged statistical mechanics sums and electrostatic energy estimation for each microstate. Summary: irGPU: Irregular strong interactions in proteins—a GPU solver is novel solution to a versatile problem in protein biophysics—atypical protonation behavior of coupled groups. The computational severity of the problem is alleviated by parallelization (via GPU kernels) which is applied for the electrostatic interaction evaluation (including explicit electrostatics via the fast multipole method) as well as statistical mechanics sums (partition function) estimation. Special attention is given to the ease of the service and encapsulation of theoretical details without sacrificing rigor of computational procedures. irGPU is not just a solution‐in‐principle but a promising practical application with potential to entice community into deeper understanding of principles governing biomolecule mechanisms. © 2015 Wiley Periodicals, Inc.